Answer:
Step-by-step explanation:
The objective is to compute that more runs are scored in games for which DH is used
Let [tex]\mu_1[/tex] denote the population mean number of runs scored for DH group
Let [tex]\mu_2[/tex] denote the population mean number of runs scored for DH group
Let [tex]n_1[/tex] and [tex]n_2[/tex] denote the sample sizes for DH and no DH
From the available information
[tex]n_1=n_2=20[/tex]
The population standard deviation of runs score is 2.54 for both the groups.
That is [tex]\sigma _1^2=\sigma_2^2=2.54[/tex]
The null hypothesis is [tex]H_0:\mu_1\leq \mu_2[/tex]
The alternative hypothesis is [tex]H_1:\mu_1>\mu_2[/tex]
Let the level of significance [tex]\alpha =0.10[/tex]
Since, the population standard deviation for both the group is known , even though the sample size is less than 30 , use z test
The test statistic is
[tex]z=\frac{\bar x_1 - \bar x_2}{\sqrt{\frac{\mu_1^2}{n_1} +\frac{\mu_2^2}{n_2} } }[/tex]
The sample mean for the DH group is computed as
[tex]\bar x_1=\frac{1}{n_1} \sum_{i=1}^{n_1}\\\\=\frac{1}{20} (0+6+8+2+2+4+7+7+6+5+1+1+5+4+4+5+7+11+10+0)\\\\=\frac{92}{20}\\\\=4.6[/tex]
The sample mean of no DH is computed as
[tex]\bar x_2=\frac{1}{n_2} \sum_{i=1}^{n_1}\\\\=\frac{1}{20} (3+6+2+4+0+5+7+6+1+8+12+4+6+3+4+0+5+2+1+4)\\\\=\frac{83}{20}\\\\=4.15[/tex]
The test statistic is
[tex]z=\frac{\bar x_1 - \bar x_2}{\sqrt{\frac{\mu_1^2}{n_1} +\frac{\mu_2^2}{n_2} } }[/tex]
[tex]=\frac{4.60-4.15}{\sqrt{\frac{2.54^2}{20} +\frac{2.54^2}{20} } } \\\\=\frac{0.45}{\sqrt{0.32258+0.32258} } \\\\=\frac{0.45}{0.803219}\\\\=0.5602[/tex]
P-Value
From the alternative hypothesis, it is clear that the test is one tailed test
P-value = P(Z > z)
[tex]1-P(Z\leq z)\\\\1-P(Z\leq 0.5602)[/tex]
1 - normsdist (0.5602)
(using excel function "normsdist(z)")
= 1 - 0.7123
= 0.2877
Therefore, the P-Value is 0.2877
Decision Rule
Reject the null hypothesis , if the p-value is less than the level of significance . that is p-value is < 0.10
Here, the p-value is 0.2877 which is greater than the level of significance 0.10
so , fail to reject the null hypothesis and conclude that there is no sufficient evidence to support the claim that more runs scored in games for which DH is used.
